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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’11 : Mathematics for Computer Science February 20 Prof. Albert R Meyer revised Sunday 20 th February, 2011, 15:11 Problem Set 3 Due : February 25 Reading: Chapter 5.1 – 5.2 , Chapter 6 Latest times for comments on different sections are indicated in the online tutor problem set TP.4. Problem 1. In this problem you will prove a fact that may surprise you —or make you even more convinced that set theory is nonsense: the half-open unit interval is actually the same size as the nonnegative quadrant of the real plane! 1 Namely, there is a bijection from .0;1Ł to OE0; 1 / 2 . (a) Describe a bijection from .0;1Ł to OE0; 1 / . Hint: 1=x almost works. (b) An infinite sequence of the decimal digits f ; 1 ;:::; 9 g will be called long if it has infinitely many occurrences of some digit other than 0. Let L be the set of all such long sequences. Describe a bijection from L to the half-open real interval .0;1Ł . Hint: Put a decimal point at the beginning of the sequence. (c) Describe a surjective function from L to L 2 that involves alternating digits from two long sequences. a Hint: The surjection need not be total....
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- Spring '11
- Computer Science, Natural number, Strong Induction, Albert R Meyer, weakly decreasing sequence, Science Prof. Albert